For the temperature structure of the flow, we use the results of [1]}, [2]}. They have found that the radiative precursor [3]}, [4]} of the shock extends to the Hill sphere (formally to infinity). This means that the infalling gas is preheated by the radiation escaping from the shock, which the work of [5]} suggests occurs mainly through the absorption of Ly \(\alpha \)  photons since they carry most of the shock emission flux. This contrasts to the stellar case, in which the precursor region is thin and located close to the star (see e.g. [6]}, [7]}, [8]}) and is due to the absorption of photoionising radiation.
This infinite precursor implies that the temperature profile is given by
\(T(r) = T_0 \left(\frac{r}{R_{\mathrm {p}}}\right)^{-1/2}. \) 
